Limit of an integral
- Tolaso J Kos
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Limit of an integral
The following exercise is just an alternative of IMC 2017/2/1 problem. It is quite easy but it's not a bad idea to have it here as well.
Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that
$$\lim_{n \rightarrow +\infty} \int_0^1 f(n x)\, {\rm d}x =0$$
Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that
$$\lim_{n \rightarrow +\infty} \int_0^1 f(n x)\, {\rm d}x =0$$
Imagination is much more important than knowledge.
Re: Limit of an integral
Since $\lim \limits_{x \rightarrow +\infty} f(x) =0$ the result follows immediately by making the change of variables $u=nx$ .
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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